Is $\mathbb{R}$ a subspace of $\mathbb{R}^2$?
Solution 1:
This is indeed an important question. No doubt that $\mathbb{R}$ is isomorphic to many subspaces of $\mathbb{R}^2$, or in other words, $\mathbb{R}$ can be embedded in many ways in $\mathbb{R}^2$. The thing is that there isn't any specific subspace in $\mathbb{R}^2$ which is the best one to represent $\mathbb{R}$.
If you want to treat $\mathbb{R}$ as a subspace, you need to specify the embedding $\mathbb{R}\hookrightarrow\mathbb{R}^2$ you refer to.
I would say that as long as we don't choose the embedding, $\mathbb{R}$ is not a subspace of $\mathbb{R}^2$.